Firstly, by [[1]}, Theorem 4 and Remarks 8, 9] (see also [[2]}, Theorem 4.2.10]), there is a projective \(S\) -scheme \(\widetilde{M}^{ss}:=\widetilde{M}^{ss}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\)  and a morphism \(\widetilde{\Theta }: \widetilde{\mathcal {R}}^{ss}\rightarrow \widetilde{M}^{ss}\)  such that \(\widetilde{\Theta }\)  is a universal good quotient for the \(\mathrm {SL}(V)\) -action. And there is an open subscheme \(\widetilde{M}^s:=\widetilde{M}^s_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\subseteq \widetilde{M}^{ss}\)  such that \(\widetilde{\mathcal {R}}^s=\widetilde{\Theta }^{-1}(\widetilde{M}^s)\)  and \(\widetilde{\Theta }:\widetilde{\mathcal {R}}^s\rightarrow \widetilde{M}^s\)  is a universal geometric quotient.
Secondly, combined with Theorem REF , it follows from the arguments in [[3]}, Lemmas 4.11, 4.12 and Remark 4.13] that for any geometric point \(s\)  of \(S\) , two points \(([a_{1}],[q_{1}])\)  and \(([a_{2}],[q_{2}])\)  in \(\widetilde{\mathcal {R}}^{ss}\)  restricted on the fiber over \(s\)  intersect if and only if their corresponding \(\delta \) -semistable pairs \((\mathcal {F}_{1},\varphi _{1})|_{s}\)  and \((\mathcal {F}_{2},\varphi _{2})|_{s}\)  are \(S\) -equivalent (see [[3]}, Definition 2.33]).
Next, we follow the similar argument in [[2]}, Lemma 4.3.1] or [[6]}, Theorem 4.12] to prove the corepresentability for the moduli functor \(\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\) . Assume that \([(\mathcal {F},\varphi )]\)  is an element of \(\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )(T)\) , i.e., there exists a flat family of \(\delta \) -(semi)stable pairs \((\mathcal {F},\varphi )\)  on \(\mathcal {X}\times _{S}T\)  with modified Hilbert polynomial \(P\)  parametrized by \(T\) . Set
\(\mathcal {W}_{\mathcal {F}}:=p_{X\times _{S}T,T*}\left(F_{p_{\mathcal {X}\times _{S}T,\mathcal {X}}^*\mathcal {E}}(\mathcal {F})\otimes p_{X\times _{S}T,X}^*\mathcal {O}_{X}(m)\right).\) 
By [[7]}, Proposition 1.5], we have \(F_{p_{\mathcal {X}\times _{S}T,\mathcal {X}}^*\mathcal {E}}(\mathcal {F})_{t}\cong F_{\check{\pi }_{t}^*\mathcal {E}}(\mathcal {F}_{t})\) , which is \(m\) -regular for each point \(t\in T\)  by the assumption of \(m\) . Then \(h^0(F_{p_{\mathcal {X}\times _{S}T,\mathcal {X}}^*\mathcal {E}}(\mathcal {F})_{t}(m))=P(m)=\dim V\) . By [[8]}, Corollary 12.9], \(\mathcal {W}_{\mathcal {F}}\)  is locally free of rank \(P(m)\) .
Let \(Fb(\mathcal {F}):=\mathrm {Isom}(V\otimes \mathcal {O}_{T},\mathcal {W}_{\mathcal {F}})\)  be the frame bundle with natural projection \(\overline{q}:Fb(\mathcal {F})\rightarrow T\)  which is a \(\mathrm {GL}(V)\) -torsor over \(T\) . Denote the universal trivialization by \(\varpi :V\otimes \mathcal {O}_{Fb(\mathcal {F})}\xrightarrow{}\overline{q}^*\mathcal {W}_{\mathcal {F}}\) . One can pull back the surjection
\(p_{X\times _{S}T,T}^*\mathcal {W}_{\mathcal {F}}\otimes p_{X\times _{S}T,X}^*\mathcal {O}_{X}(-m)\rightarrow F_{p_{\mathcal {X}\times _{S}T,\mathcal {X}}^*\mathcal {E}}(\mathcal {F})\)  by the map \(\mathrm {id}_{X}\times \bar{q}:X\times _{S}Fb(\mathcal {F})\rightarrow X\times _{S}T\) , and then apply the functor \(G_{p_{\mathcal {X}\times _{S}Fb(\mathcal {F})}^*\mathcal {E}}\)  and the surjective map \(\theta _{p_{\mathcal {X}\times _{S}Fb(\mathcal {F})}^*\mathcal {E}}((\mathrm {id}_{\mathcal {X}}\times \overline{q})^*\mathcal {F})\)  to obtain the quotient
\(p_{\mathcal {X}\times _{S} Fb(\mathcal {F}),\mathcal {X}}^*(V\otimes \mathcal {E}\otimes \pi ^*\mathcal {O}_{X}(-m))\rightarrow (\mathrm {id}_{\mathcal {X}}\times \overline{q})^*\mathcal {F}.\) 
This implies that
one has a morphism \(Fb(\mathcal {F})\rightarrow \widetilde{\mathcal {Q}}\)  due to the universal property of \(\widetilde{\mathcal {Q}}\) . On the other hand, from the morphism \(\varphi : \pi _{\mathcal {X}/S}^*\mathcal {F}_{0}\rightarrow \mathcal {F}\)  (notice that \(\pi _{\mathcal {X}/S}=p_{\mathcal {X}\times _{S}T,\mathcal {X}}\) ), we have
\(p_{X\times _{S}T,T*}(F_{p_{\mathcal {X}\times _{S}T,\mathcal {X}}^*\mathcal {E}}(\varphi )\otimes p_{X\times _{S}T,X}^*\mathcal {O}_{X}(m)):H^0(F_{\mathcal {E}}(\mathcal {F}_{0})(m))\otimes \mathcal {O}_{T}\rightarrow \mathcal {W}_{\mathcal {F}}\) 
where we use the isomorphism \(p_{X\times _{S}T,X}^*F_{\mathcal {E}}(\mathcal {F}_{0})\cong F_{\pi _{\mathcal {X}/S}^*\mathcal {E}}(\pi _{\mathcal {X}/S}^*\mathcal {F}_{0})\) , and then pull back the above morphism by \(\overline{q}\)  and compose with \(\varpi ^{-1}\) , we obtain
\(H^0(F_{\mathcal {E}}(\mathcal {F}_{0})(m))\otimes \mathcal {O}_{Fb(\mathcal {F})}\rightarrow V\otimes \mathcal {O}_{Fb(\mathcal {F})}.\) 
Then we also have a morphism \(Fb(\mathcal {F})\rightarrow \mathbb {P}\)  by the universal property of \(\mathbb {P}\) . By the definition of \(\mathcal {Z}^\prime \)  and Theorem REF , we have the morphism \(Fb(\mathcal {F})\rightarrow \mathbb {P}\times _{S}\widetilde{\mathcal {Q}}\)  factoring through
a morphism \(\phi : Fb(\mathcal {F})\rightarrow \widetilde{\mathcal {R}}^{(s)s}\)  which is obviously \(\mathrm {GL}(V)\) -equivariant. Set \(\widetilde{\mathfrak {R}}^{(s)s}:=[\widetilde{\mathcal {R}}^{(s)s}/\mathrm {GL}(V)]\) . As in [[6]}, Theorem 4.12] (see alternatively [[2]}, Lemma 4.3.1]), we have a natural transformation
\(\Upsilon :\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\rightarrow [\widetilde{\mathfrak {R}}^{(s)s}]\) 
where \([\widetilde{\mathfrak {R}}^{(s)s}]\)  is the contravariant functor such that for any \(S\) -scheme \(T\)  of finite type, \([\widetilde{\mathfrak {R}}^{(s)s}](T)\)  is the set of isomorphism classes of objects of \(\widetilde{\mathfrak {R}}^{(s)s}(T)\) . In fact, one can show that two functors \(\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\)  and \([\widetilde{\mathfrak {R}}^{(s)s}]\)  are isomorphic by following the similar argument in [[6]}, Theorem 4.12].
By [[12]}, Theorem 13.6], we have the morphism \([\widetilde{\mathcal {R}}^{(s)s}/\mathrm {SL}(V)]\rightarrow \widetilde{M}^{(s)s}:=\widetilde{\mathcal {R}}^{(s)s}{\tt //}\mathrm {SL}(V)\) , which is a good moduli space on the fiber over each geometric point \(s\)  of \(S\) . And we have the morphism \(\widetilde{\mathfrak {R}}^{(s)s}\rightarrow \widetilde{M}^{(s)s}\)  by the discussion following [[6]}, Proposition 4.11] (see also the proof of [[14]}, Theorem 1.21-(1)]), which factors through \(\widetilde{\mathfrak {R}}^{(s)s}\rightarrow [\widetilde{\mathfrak {R}}^{(s)s}]\) . Composing \(\Upsilon \)  with the map \([\widetilde{\mathfrak {R}}^{(s)s}]\rightarrow \widetilde{M}^{(s)s}\) , we have the natural transformation \(\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\rightarrow \mathrm {Mor}(-,\widetilde{M}^{(s)s})\)  by the Yoneda Lemma. Suppose there is a natural transformation \(\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\rightarrow \mathrm {Mor}(-,\overline{M})\) , where \(\overline{M}\)  is a \(S\) -scheme. Since the universal family \((\check{\mathcal {F}}_{\mathcal {X}\times _{S}\widetilde{\mathcal {R}}^{(s)s}},\check{\varphi }_{\mathcal {X}\times _{S}\widetilde{\mathcal {R}}^{(s)s}})\)  is an element in \(\mathcal {M}^{ss}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )(\widetilde{\mathcal {R}}^{(s)s})\) , then we have a morphism \(\widetilde{\mathcal {R}}^{(s)s}\rightarrow \overline{M}\)  which is clearly \(\mathrm {SL}(V)\) -equivariant with the trivial \(\mathrm {SL}(V)\) -action on \(\overline{M}\) . Since \(\widetilde{\mathcal {R}}^{(s)s}\rightarrow \widetilde{M}^{(s)s}\)  is a categorical quotient, we have the following commutative diagram
\({\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )[dr] [r] & \mathrm {Mor}(-,\widetilde{M}^{(s)s})[d]\\&\mathrm {Mor}(-,\overline{M})}\) 
where the morphim \(\widetilde{M}^{(s)s}\rightarrow \overline{M}\)  is unique.
By [[2]}, Definition 2.2.1], this implies that \(\widetilde{M}^{(s)s}\)  corepresents the functor \(\mathcal {M}^{(s)s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\) .
Therefore, the proof follows from the above results.
